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Theorem nelbrnel 46703
Description: A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
Assertion
Ref Expression
nelbrnel ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))

Proof of Theorem nelbrnel
StepHypRef Expression
1 nelbr 46701 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
2 df-nel 3044 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
31, 2bitr4di 288 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2098  wnel 3043   class class class wbr 5152   _∉ cnelbr 46698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nel 3044  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-nelbr 46699
This theorem is referenced by: (None)
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